# NAG CL Interfaces19abc (kelvin_​bei)

## 1Purpose

s19abc returns a value for the Kelvin function $\mathrm{bei}x$.

## 2Specification

 #include
 double s19abc (double x, NagError *fail)
The function may be called by the names: s19abc, nag_specfun_kelvin_bei or nag_kelvin_bei.

## 3Description

s19abc evaluates an approximation to the Kelvin function $\mathrm{bei}x$.
Note:  $\mathrm{bei}\left(-x\right)=\mathrm{bei}x$, so the approximation need only consider $x\ge 0.0$.
The function is based on several Chebyshev expansions:
For $0\le x\le 5$,
 $bei⁡x = x24 ∑ r=0 ′ ar Tr t , with ​ t=2 x5 4 - 1 ;$
For $x>5$,
 $bei⁡x=ex/22πx 1+1xat sin⁡α-1xbtcos⁡α$
 $+ex/22π x 1+1xct cos⁡β-1xdtsin⁡β$
where $\alpha =\frac{x}{\sqrt{2}}-\frac{\pi }{8}$, $\beta =\frac{x}{\sqrt{2}}+\frac{\pi }{8}$,
and $a\left(t\right)$, $b\left(t\right)$, $c\left(t\right)$, and $d\left(t\right)$ are expansions in the variable $t=\frac{10}{x}-1$.
When $x$ is sufficiently close to zero, the result is computed as $\mathrm{bei}x=\frac{{x}^{2}}{4}$. If this result would underflow, the result returned is $\mathrm{bei}x=0.0$.
For large $x$, there is a danger of the result being totally inaccurate, as the error amplification factor grows in an essentially exponential manner; therefore the function must fail.

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{x}$double Input
On entry: the argument $x$ of the function.
2: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL_ARG_GT
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{x}}\right|\le 〈\mathit{\text{value}}〉$.
$\left|{\mathbf{x}}\right|$ is too large for an accurate result to be returned and the function returns zero.

## 7Accuracy

Since the function is oscillatory, the absolute error rather than the relative error is important. Let $E$ be the absolute error in the function, and $\delta$ be the relative error in the argument. If $\delta$ is somewhat larger than the machine precision, then we have:
 $E≃ x2 - ber1⁡x+ bei1⁡x δ$
(provided $E$ is within machine bounds).
For small $x$ the error amplification is insignificant and thus the absolute error is effectively bounded by the machine precision.
For medium and large $x$, the error behaviour is oscillatory and its amplitude grows like $\sqrt{\frac{x}{2\pi }}{e}^{x/\sqrt{2}}$. Therefore it is impossible to calculate the functions with any accuracy when $\sqrt{x}{e}^{x/\sqrt{2}}>\frac{\sqrt{2\pi }}{\delta }$. Note that this value of $x$ is much smaller than the minimum value of $x$ for which the function overflows.

## 8Parallelism and Performance

s19abc is not threaded in any implementation.

None.

## 10Example

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.

### 10.1Program Text

Program Text (s19abce.c)

### 10.2Program Data

Program Data (s19abce.d)

### 10.3Program Results

Program Results (s19abce.r)